Problem 40 - Intermediate Algebra

Question

Determine whether each equation defines \(y\) to be a function of \(x .\) If it does not, find two ordered pairs where more than one value of \(y\) corresponds to a single value of \(x .\) $$ x=|y| $$

Step-by-Step Solution

Answer

The equation does not define \( y \) as a function of \( x \). For example, when \( x = 1 \), \( y \) can be \( 1 \) or \( -1 \).

Step 1: Understanding the Problem

We need to determine if the equation \( x = |y| \) defines \( y \) as a function of \( x \). This means for every \( x \) there should be only one \( y \).

Step 2: Analyzing the Absolute Value Equation

The equation \( x = |y| \) means \( y \) can be either \( x \) or \(-x\). Both of these are valid for any non-negative \( x \).

Step 3: Checking for Function Criteria

For a relation to be a function, each \( x \) should correspond to exactly one \( y \). Here, for any positive \( x \), there are two corresponding \( y \) values, \( x \) and \(-x\).

Step 4: Counterexamples

Take \( x = 1 \), then the equation gives two possible values for \( y \) which are \( y = 1 \) and \( y = -1 \). Thus, \( (1, 1) \) and \( (1, -1) \) are two ordered pairs for the same \( x \).

Step 5: Conclusion

Since there are two \( y \) values for a single \( x \), the equation \( x = |y| \) does not define \( y \) as a function of \( x \).

Key Concepts

Absolute Value

The concept of absolute value is fundamental in mathematics. It refers to the non-negative value of a number without regard to its sign. In other words, the absolute value of a number is how far it is from zero, regardless of direction. For example:

• The absolute value of 5 is 5, because it is 5 units from zero.
• Similarly, the absolute value of -5 is also 5, because it is 5 units from zero, but in the opposite direction.

Mathematically, the absolute value of a number, say \( a \), is denoted as \( |a| \). Therefore:

• \( |a| = a \) if \( a \geq 0 \)
• \( |a| = -a \) if \( a < 0 \)

In the context of the exercise, the equation \( x = |y| \) implies that \( y \) can take on values of both \( x \) and \(-x\), due to the nature of absolute value. This is a key detail when examining if equations like this define a function or not.

Ordered Pairs

Ordered pairs are an essential concept in mapping relations in mathematics. An ordered pair is a pair of elements grouped together in a specific order, typically represented as \((x, y)\). The first element is often referred to as the "input" or "independent variable," while the second element is the "output" or "dependent variable." Here are a few key points:

• Order matters: \((x, y)\) is different from \((y, x)\).
• Each ordered pair represents a potential mapping of \( x \) to \( y \).
• Visual graphs of functions often use these pairs to plot points.

In relation to our exercise, the problem requires determining if a specific value of \( x \) corresponds to more than one \( y \) value. The example given (\( (1, 1) \) and \( (1, -1) \)) illustrates that more than one ordered pair can be formed with the same \( x \) value, showing that \( x = |y| \) does not meet the function criteria.

Function Criteria

To determine whether an equation defines a function, we need to check the function criteria: every \( x \) must be paired with only one \( y \). This uniqueness is what makes a relationship a function.

• If you can find any \( x \) that maps to more than one \( y \), the relation is not a function.
• A simple test is the vertical line test for graphs: if a vertical line intersects the graph in more than one place, it is not a function.

In the context of the given equation \( x = |y| \):

• For every positive \( x \), there are two possible corresponding \( y \) values, namely \( x \) and \(-x\).
• Hence, \( (1, 1) \) and \( (1, -1) \) as potential pairs violate the function rule of uniqueness.

This lack of a one-to-one correspondence between \( x \) and \( y \) means \( x = |y| \) does not satisfy the criteria needed to qualify as a function.